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Name:  
Miloslav Znojil  
Reviewer number:  
9689  
Email:  
znojil@ujf.cas.cz  
Item's zblNumber:  
DE 0162 4986 3  
Author(s):  
Grebert, B; Kappeler, T.  
Shorttitle:  
Estimates of eigenvalues for Zakharov Shabat system  
Source:  
Asymptotic Anal. 25, No. 34, 201237 (2001)  
Classification:  
Primary Classification:  
 
Secondary Classification:  
 
Keywords:  
complex version of the ZakharovShabat system; periodic and Dirichlet boundary conditions; asymptotic distribution of eigenvalues;  
Review:  
Nonlinear Schr\"{o}dinger equation may be assigned (or represented and analyzed in terms of) a Lax pair of operators of the Zakharov Shabat (in essence: perturbed Dirac operator) type. This context and related specific range of applications limit, often, the assumptions and analysis, typically, to the selfadjoint case. The paper deals with the generalized nonselfadjoint twobytwo complex Zakharov Shabat linear differential operator, with (in the Diracequation language) ``potential" term assumed to lie within a certain (suitably weighted) Sobolev space. \par The spectrum is still known to be discrete (for both the periodic and Dirichlet boundary conditions). The suitable asymptotic expansions of the eigenvalues are available in the selfadjoint case but not in its present generalization. The authors employ a nonstandard method (called a LyapunovSchmidt type decomposition in the paper) and offer the two basic theorems on the asymptotic distribution of the (complex) eigenvalues. \par The text is inspired by the recent similar analysis of the linear Schroedinger operators. This puts this paper in a perspective of a natural methodical development since, in the later context, the method is well known under many other names (e.g., as a Loewdin's projection operator method in perturbation theory and in its applications in quantum chemistry or as the feshbach's effective operator method in quantum mechanics and nuclear physics, etc). Its present application is innovative and leads to a surprisingly compact picture of the generalized case, with the key idea lying in a successful elimination of all the ``irrelevant" Fourier components of the eigenfunctions so that we are left with the twobytwo ``effective" algebraic eigenvalue problem.  
Remarks to the editors:  